STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
GRANULARITY ADJUSTMENT FOR
DYNAMIC MULTIPLE FACTOR MODELS :
SYSTEMATIC VS UNSYSTEMATIC RISKS
Patrick GAGLIARDINI and Christian GOURIÉROUX
Patrick Gagliardini and Christian Gouriéroux
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STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
INTRODUCTION
Risk measures such as
Value-at-Risk (VaR)
Expected Shortfall (also called TailVaR)
Distortion Risk Measures (DRM)
are the basis of the new risk management policies and
regulations for Finance (Basel 2) and Insurance (Solvency 2)
Patrick Gagliardini and Christian Gouriéroux
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Risk measures are used to
i) define the reserves (minimum required capital)
needed to hedge risky investments
(Pillar 1 of Basel 2 regulation)
ii) monitor the risk by means of internal risk models
(Pillar 2 of Basel 2 regulation)
Patrick Gagliardini and Christian Gouriéroux
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Risk measures have to be computed for large portfolios of
individual contracts :
portfolios of loans and mortgages
portfolios of life insurance contracts
portfolios of Credit Default Swaps (CDS)
and for derivative assets written on such large portfolios :
Mortgage Backed Securities (MBS)
Collateralized Debt Obligations (CDO)
Derivatives on iTraxx
Insurance Linked Securities (ILS) and longevity bonds
Patrick Gagliardini and Christian Gouriéroux
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The value of portfolio risk measures may be difficult to compute
even numerically, due to
i) the large size of the portfolio (between 100 and
10, 000 − 100, 000 contracts)
ii) the nonlinearity of risks such as default, loss given
default, claim occurrence, prepayment, lapse
iii) the dependence between individual risks, which is
induced by the systematic risk components
Patrick Gagliardini and Christian Gouriéroux
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CONCLUDING REMARKS
The granularity principle [Gordy (2003)] allows to :
derive closed form expressions for portfolio risk measures
at order 1/n, where n denotes the portfolio size
separate the effect of systematic and idiosyncratic risks
[Gouriéroux, Laurent, Scaillet (2000), Tasche (2000), Wilde
(2001), Martin, Wilde (2002), Emmer, Tasche (2005), Gordy,
Lutkebohmert (2007)]
The value of the portfolio risk measure RM n is decomposed as
RMn = Asymptotic risk measure (corresponding to n = ∞)
1
+ Adjustment term
n
Patrick Gagliardini and Christian Gouriéroux
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CONCLUDING REMARKS
The asymptotic portfolio risk measure, called
Cross-Sectional Asymptotic (CSA) risk measure
captures the effect of systematic risk on the portfolio value
The adjustment term, called
Granularity Adjustment (GA)
captures the effect of idiosyncratic risks which are not fully
diversified for a portfolio of finite size
Patrick Gagliardini and Christian Gouriéroux
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STATIC MULTIPLE RISK FACTOR MODEL
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CONCLUDING REMARKS
WHAT IS THIS PAPER ABOUT ?
We derive the granularity adjustment of Value-at-Risk (VaR) for
general risk factor models where the systematic factor can be
multidimensional and
dynamic
We apply the GA approach to compute the portfolio VaR in a
dynamic model with stochastic default and loss given default
Patrick Gagliardini and Christian Gouriéroux
Granularity for Risk Measures
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
Outline
1
STATIC MULTIPLE RISK FACTOR MODEL
Homogenous Portfolio
Portfolio Risk
Asymptotic Portfolio Risk
Granularity Principle
2
EXAMPLES
3
DYNAMIC RISK FACTOR MODEL
4
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
5
CONCLUDING REMARKS
Patrick Gagliardini and Christian Gouriéroux
Granularity for Risk Measures
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
1. STATIC MULTIPLE RISK FACTOR MODEL
Patrick Gagliardini and Christian Gouriéroux
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STATIC MULTIPLE RISK FACTOR MODEL
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CONCLUDING REMARKS
1.1 Homogenous Portfolio
The individual risks
yi = c(F , ui )
depend on the vector of systematic factors F and the
idiosyncratic risks ui
Distributional assumptions
A.1 : F and (u1 , . . . , un ) are independent
A.2 : u1 , . . . , un are independent, identically distributed
The portfolio is homogenous since the individual risks are
exchangeable
Patrick Gagliardini and Christian Gouriéroux
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CONCLUDING REMARKS
Example 1 : Value of the Firm model [Vasicek (1991)]
The risk variables yi are default indicators
⎧
⎨ 1, if Ai < Li (default)
yi =
⎩
0, otherwise
where Ai and Li are asset value and liability [Merton (1974)]
The log asset/liability ratios are such that
log (Ai /Li ) = F + ui
Thus we get the single-factor model
yi = 1lF + u < 0
i
considered in Basel 2 regulation [BCBS (2001)]
Patrick Gagliardini and Christian Gouriéroux
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Example 2 : Model with Stochastic Drift and Volatility
The risks are (opposite) asset returns
yi = F1 + (F2 )1/2 ui
where factor F = (F1 , F2 ) is bivariate and includes
common stochastic drift F1
common stochastic volatility F2
Patrick Gagliardini and Christian Gouriéroux
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1.2 Portfolio Risk
The total portfolio risk is :
Wn =
n
i=1
yi =
n
c(F , ui )
i=1
and corresponds to either a Profit and Loss (P&L) or a Loss
and Profit (L&P) variable
The distribution of the portfolio risk Wn is typically unknown in
closed form due to risk dependence and aggregation
Numerical integration or Monte-Carlo simulation can be very
time consuming
Patrick Gagliardini and Christian Gouriéroux
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1.3 Asymptotic Portfolio Risk
Limit theorems such as the Law of Large Numbers (LLN) and
the Central Limit Theorem (CLT) cannot be applied to the
sequence y1 , . . . , yn due to the common factors
However, LLN and CLT can be applied conditionally on factor
values !
This is the so-called condition of infinitely fine grained
portfolio in Basel 2 terminology
Patrick Gagliardini and Christian Gouriéroux
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By applying the CLT conditionally on factor F , for large n we
have
X
Wn /n = m(F ) + σ(F ) √ + O(1/n)
n
where
m(F ) = E[yi |F ] is the conditional individual expected risk
σ 2 (F ) = V [yi |F ] is the conditional individual volatility
X is a standard Gaussian variable independent of F
the term at order O(1/n) is conditionally zero mean
Patrick Gagliardini and Christian Gouriéroux
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1.4 Granularity Principle
i) Standardized risk measures
The VaR of the portfolio explodes when portfolio size n → ∞
It is preferable to consider the VaR per individual asset included
in the portfolio, that is the quantile of W n /n
For a L&P variable the VaR at level α is defined by the condition
P[Wn /n < VaRn (α)] = α
where α = 95%, 99%, 99.5%
Patrick Gagliardini and Christian Gouriéroux
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ii) The CSA risk measure
A portfolio with infinite size n = ∞ is not riskfree since the
systematic risks are undiversifiable !
In fact, for n = ∞ we have :
Wn /n = m(F )
which is stochastic
We deduce that the CSA risk measure VaR ∞(α) is the quantile
associated with the systematic component m(F ) :
P[m(F ) < VaR∞ (α)] = α
[Vasicek (1991)]
Patrick Gagliardini and Christian Gouriéroux
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iii) Granularity Adjustment for the risk measure
The main result in granularity theory applied to risk measures
provides the next term in the asymptotic expansion of VaR n (α)
with respect to n in a neighbourhood of n = ∞
Theorem 1 : We have
VaRn (α) = VaR∞(α) +
where
1
GA(α) + o(1/n)
n
d log g∞ (w)
E[σ 2 (F )|m(F ) = w]
dw
d
2
E[σ (F )|m(F ) = w]
+
dw
w = VaR∞ (α)
1
GA(α) = −
2
and g∞ denotes the probability density function of m(F )
Patrick Gagliardini and Christian Gouriéroux
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The expansion in Theorem 1 is useful since VaR ∞(α) and
GA(α) do not involve large dimensional integrals !
The second term in the expansion is of order 1/n. Hence,
the granularity approximation can be accurate, even for
rather small values of n ( 100)
For single-factor models Theorem 1 provides the
granularity adjustment derived in Gordy (2003)
Theorem 1 applies for general multi-factor models
The expansion is easily extended to the other Distortion
Risk Measures [Wang (1996, 2000)], which are weighted
averages of VaR, in particular to the Expected Shortfall
Patrick Gagliardini and Christian Gouriéroux
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CONCLUDING REMARKS
2. EXAMPLES
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Example 1 : Value of the firm model
√
yi = 1l
−Φ−1 (PD) + ρF ∗ + 1 − ρui∗ < 0
where F ∗ , ui∗ ∼ N(0, 1) and PD is the unconditional probability
of default and ρ is the asset correlation
√
Φ−1 (PD) + ρΦ−1 (α)
√
VaR∞(α) = Φ
1−ρ
⎧
1 − ρ −1
⎪
⎪
Φ (α) − Φ−1 [VaR∞(α)]
1⎨
ρ
VaR∞(α)[1 − VaR∞(α)]
GA(α) =
2⎪
φ Φ−1 [VaR∞(α)]
⎪
⎩
+2VaR∞(α) − 1
[cf. Emmer, Tasche (2005), formula (2.17)]
Patrick Gagliardini and Christian Gouriéroux
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1
n GA(0.99)
V aR∞ (0.99)
1
0.03
PD=0.5%
PD=1%
PD=5%
PD=20%
0.9
0.025
0.8
0.7
0.02
0.6
0.015
0.5
0.4
0.01
0.3
0.2
0.005
0.1
0
0
0.2
0.4
Patrick Gagliardini and Christian Gouriéroux ()
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0.6
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Granularity for Risk Measures
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0.6
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n GA(0.99)
10
0.5
8
0.4
6
0.3
ρ=0.05
ρ=0.12
ρ=0.24
ρ=0.50
0.2
0.1
0
0
x 10
0.2
0.4
PD
Patrick Gagliardini and Christian Gouriéroux ()
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0.8
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0
0.2
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PD
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n GA(0.99)
−3
20
x 10
ρ=0.05
ρ=0.12
ρ=0.24
ρ=0.50
18
16
14
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10
8
6
4
2
0
0
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0.4
0.6
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STATIC MULTIPLE RISK FACTOR MODEL
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CONCLUDING REMARKS
Heterogeneity can be introduced into the model by including
multiple idiosyncratic risks
Value of the firm model with heterogenous loadings
√
= c(F ∗ , ui )
yi = 1l
−Φ−1 (PD) + ρi F ∗ + 1 − ρi vi < 0
where ui = (vi , ρi ) includes both firm specific shocks and factor
loadings
Portfolio with heterogenous exposures
Wn =
n
i=1
Ai yi =
n
∗
Ai c(F , ui ) =
i=1
n
c(F ∗ , wi )
i=1
where Ai are the individual exposures and w i = (ui , Ai )
Patrick Gagliardini and Christian Gouriéroux
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Example 2 : Stochastic Drift and Volatility
yi ∼ N(F1 , exp F2 )
μ1
ρσ1 σ2
σ12
,
where F = (F1 , F2 ) ∼ N
μ2
ρσ1 σ2
σ22
We have m(F ) = F 1 , σ 2 (F ) = exp F2 and
ρσ2
d
log E[σ 2 (F )|m(F ) = w] =
= leverage effect !
dw
σ1
We deduce that :
VaR∞(α) = μ1 + σ1 Φ−1 (α)
ρ2 σ22
v22 −1
−1
[Φ (α) − ρσ2 ] exp ρσ2 Φ (α) −
GA(α) =
2σ1
2
where v22 = E[exp F2 ] = exp μ2 + σ22 /2
Patrick Gagliardini and Christian Gouriéroux
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Example 3 : Stochastic Probability of Default and Loss
Given Default
A zero-coupon corporate bond with loss at maturity :
yi = Zi LGDi
where Zi is the default indicator and LGD i is the Loss Given
Default
Conditional on factor F = (F 1 , F2 ) , variables Zi and LGDi are
independent such that
Zi ∼ B(1, F1 ),
LGDi ∼ Beta(a(F2 ), b(F2 ))
and
E [LGDi |F ] = F2 , V [LGDi |F ] = γF2 (1 − F2 )
with γ ∈ (0, 1) constant
Patrick Gagliardini and Christian Gouriéroux
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Example 3 : Stochastic Probability of Default and Loss
Given Default (cont.)
We get a two-factor model
F1 = P[Zi |F ] is the conditional Probability of Default
F2 = E[LGDi |F ] is the conditional Expected Loss Given Default
The two factors F1 and F2 can be correlated
We derive the CSA risk measure and the GA with
m(F ) = F1 F2
σ 2 (F ) = γF2 (1 − F2 )F1 + F1 (1 − F1 )F22
Patrick Gagliardini and Christian Gouriéroux
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CONCLUDING REMARKS
3. DYNAMIC RISK FACTOR MODEL
Patrick Gagliardini and Christian Gouriéroux
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3.1 The model
Past observations are informative about future risk !
Consider a dynamic framework where the factor values include
all relevant information
Static relationship between individual risks and systematic
factors
yi,t = c(Ft , ui,t )
A.3 : The (ui,t ) are iid and independent of (F t )
(Ft ) is a Markov stochastic process
The dynamics of individual risks are entirely due to the
underlying dynamic of the systematic risk factor
Patrick Gagliardini and Christian Gouriéroux
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3.2 Standardized portfolio risk measure
Future portfolio risk per individual asset
1
yi,t+1
n
n
Wn,t+1 /n =
i=1
The dynamic VaR is defined by the equation :
P[Wn,t+1 /n < VaRn,t (α)|In,t ] = α
where information In,t includes all current and past individual
risks yi,t , yi,t−1 , . . ., for i = 1, . . . , n, but not the factor values
The quantile VaR n,t (α) depends on the date t through the
information In,t
Patrick Gagliardini and Christian Gouriéroux
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3.3 Granularity adjustment
i) A first granularity adjustment
The general theory can be applied conditionally on the current
factor value Ft . The conditional VaR is defined by :
P[Wn,t+1 /n < VaRn (α, Ft )|Ft ] = α
and we have :
1
GA(α, Ft ) + o(1/n)
n
where VaR∞(α, Ft ) and GA(α, Ft ) are computed as in the static
case, but with an additional conditioning with respect to F t
VaRn (α, Ft ) = VaR∞(α, Ft ) +
Patrick Gagliardini and Christian Gouriéroux
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ii) A second granularity adjustment
However, the expansion above cannot be used directly since
the current factor value is not observable !
Let the conditional pdf of y it given Ft be denoted h(y it |ft )
The cross-sectional maximum likelihood estimator of f t
f̂nt = arg max
ft
n
log h(yit |ft )
i=1
provides a consistent approximation of ft as n → ∞
Patrick Gagliardini and Christian Gouriéroux
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Replacing ft by f̂n,t introduces an approximation error of order
1/n that requires an additional granularity adjustment
Use the approximate filtering distribution of F t given In,t at order
1/n derived in Gagliardini, Gouriéroux (2009) to get
VaRn (α) = VaR∞(α, f̂n,t ) +
1
1
GA(α, f̂n,t ) + GAfilt (α) + o(1/n)
n
n
1
GAfilt (α) is an additional granularity adjustment of the
n
risk measure due to non observability of the systematic factor
Term
GAfilt (α) is given in closed form in the paper
Patrick Gagliardini and Christian Gouriéroux
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4. DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN
DEFAULT
Patrick Gagliardini and Christian Gouriéroux
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A Value of the Firm model [Merton (1974), Vasicek (1991)] with
non-zero recovery rate and dynamic systematic factor
Risk variable is percentage loss of debt holder at time t :
yi,t = 1lAi,t <Li,t
1−
Ai,t
Li,t
=
1−
Ai,t
Li,t
+
where Ai,t and Li,t are asset value and liability at t
The loss Li,t yi,t is the payoff of a put option written on the asset
value and with strike equal to liability
Patrick Gagliardini and Christian Gouriéroux
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The log asset/liability ratios follow a linear single factor model :
Ai,t
= Ft + σui,t
log
Li,t
where (ui,t ) ∼ IIN(0, 1) and (Ft ) are independent
The systematic risk factor Ft follows a stationary Gaussian
AR(1) process :
Ft = μ + γ(Ft−1 − μ) + η 1 − γ 2 εt ,
where (εt ) ∼ IIN(0, 1) and |γ| < 1
Patrick Gagliardini and Christian Gouriéroux
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The model is parameterized by 4 structural parameters μ, η, σ
and γ, or equivalently by means of :
Ai,t
PD = P log
< 0 probability of default
Li,t
Ai,t Ai,t
ELGD = E 1 −
|
< 1 expected loss given default
Li,t Li,t
Aj,t
Ai,t
ρ = corr log
, log
asset correlation (i = j)
Li,t
Lj,t
γ first-order autocorrelation of the factor
The parameterization by PD, ELGD, ρ, γ is convenient for
calibration !
Patrick Gagliardini and Christian Gouriéroux
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The cross-sectional maximum likelihood approximation of the
factor value at date t is given by :
⎫
⎧
⎬
⎨ 1 2
log(1 − yi,t ) − ft + (n − nt ) log Φ(ft /σ)
f̂n,t = arg max − 2
⎭
ft ⎩ 2σ
i:yi,t >0
where nt =
n
1lyi,t >0 denotes the number of defaults at date t
i=1
i.e. a Tobit Gaussian cross-sectional regression !
The filtering distribution depends on the available information
through the current and lagged factor approximations f̂n,t and
f̂n,t−1 and the default frequency n t /n
Patrick Gagliardini and Christian Gouriéroux
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Parameters: PD = 5%, ELGD = 0.45, ρ = 0.12, γ = 0.5. VaR: α = 99.5%
CSA VaR and GA VaR
1.7
CSA
GA n = 100
0.2
GA n = 1000 1.6
0.18
GAfilt (α)
GArisk (α)
0.22
6
5
4
1.5
0.16
3
0.14
1.4
2
1.3
1
0.12
0.1
0
0.08
1.2
−1
0.06
1.1
−2
0.04
0.02
0
2
fˆn,t
Patrick Gagliardini and Christian Gouriéroux ()
4
1
0
2
fˆn,t
Granularity for Risk Measures
4
−3
0
2
fˆn,t
4
April 12, 2010
4/7
Parameters: PD = 1.5%, ELGD = 0.45, ρ = 0.12, γ = 0.5. VaR: α = 99.5%
GAf ilt (α)
GArisk (α)
CSA VaR and GA VaR
1.5
0.1
8
0.09
1.4
6
0.08
1.3
0.07
4
0.06
1.2
0.05
2
1.1
CSA
0.04
GA n = 100
0.03
GA
n = 1000
0
1
0.02
−2
0.9
0.01
0
0
5
fˆn,t
Patrick Gagliardini and Christian Gouriéroux ()
10
0.8
0
5
fˆn,t
Granularity for Risk Measures
10
−4
0
5
fˆn,t
April 12, 2010
10
5/7
Parameters: PD = 5%, ELGD = 0.45, ρ = 0.12, γ = 0.5. Portfolio: n = 100
Time series of default frequency nt /n and percentage portfolio loss Wn,t /n
0.25
default frequency nt /n
portfolio loss Wn,t /n
0.2
0.15
0.1
0.05
0
0
10
20
30
40
50
t
60
70
80
90
100
90
100
Time series of factor ft and factor approximation fˆn,t
6
factor ft
5
approximation fˆn,t
4
3
2
1
0
0
10
20
Patrick Gagliardini and Christian Gouriéroux ()
30
40
50
t
Granularity for Risk Measures
60
70
80
April 12, 2010
6/7
Parameters: PD = 5%, ELGD = 0.45, ρ = 0.12, γ = 0.5. Portfolio: n = 100
Time series of CSA VaR and GA VaR
0.25
CSA VaR
GA VaR n = 100
0.2
0.15
0.1
0.05
0
10
20
30
40
50
t
60
70
80
90
100
90
100
Time series of GArisk (α) and GAfilt (α)
6
GArisk
GAf ilt
4
2
0
−2
−4
0
10
20
Patrick Gagliardini and Christian Gouriéroux ()
30
40
50
t
Granularity for Risk Measures
60
70
80
April 12, 2010
7/7
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
Backtesting of CSA VaR and GA VaR
Ht = 1lWn,t /n≥VaRn,t−1 (α) − α
E [Ht ]
Corr (Ht , Ht−1 )
Corr (Ht , Ht−2 )
λ2
Corr Ht , f̂n,t−1
Corr Ht , f̂n,t−2
Corr Ht , wn,t−1 Corr Ht , wn,t−2
Patrick Gagliardini and Christian Gouriéroux
CSA
0.008
−0.007
0.002
−0.007
GA
−0.001
−0.004
−0.000
0.002
0.054
−0.022
0.005
0.002
−0.034
−0.002
0.019
0.002
Granularity for Risk Measures
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
5. CONCLUDING REMARKS
Patrick Gagliardini and Christian Gouriéroux
Granularity for Risk Measures
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
For large homogenous portfolios, closed form expressions of
the VaR and other distortion risk measures can be derived at
order 1/n
Results apply for a rather general class of risk models with
multiple factors in a dynamic framework
Two granularity adjustments are required :
The first GA concerns the conditional VaR with current factor
value assumed to be observed
The second GA accounts for the unobservability of the factor
Patrick Gagliardini and Christian Gouriéroux
Granularity for Risk Measures
STATIC MULTIPLE RISK FACTOR MODEL
EXAMPLES
DYNAMIC RISK FACTOR MODEL
DYNAMIC MODEL FOR DEFAULT AND LOSS GIVEN DEFAULT
CONCLUDING REMARKS
The GA principle appeared in Pillar 1 of the Basel Accord in
2001, concerning minimum required capital
It has been moved to Pillar 2 in the most recent version of the
Basel Accord in 2003, concerning internal risk models
The recent financial crisis has shown the importance of
distinguishing between idiosyncratic and systematic risks when
computing reserves !
The GA technology can be useful for this purpose, e.g. by
allowing to
fix different risk levels for CSA and GA VaR
smooth differently these components over the cycle when
computing the reserves
Patrick Gagliardini and Christian Gouriéroux
Granularity for Risk Measures